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## Rationality of Varieties

Birkhäuser, 2021.

Under the general editorship: G. Farkas, G. van der Geer, M. Shen, L. Taelman

Amerik E., Verbitsky M., , in : Rationality of Varieties. : Birkhäuser, 2021. P. 75-96.

Added: April 6, 2022

Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We prove that Q-Fano threefolds of Fano index ≥8 are rational. ...

Added: June 8, 2019

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1411.6570.

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent over $\boldsymbol k$ rational functions in the structure constants. ...

Added: November 25, 2014

Prokhorov Y., Rendiconti del Circolo Matematico di Palermo 2023 Vol. 2 No. 72 P. 1797-1821

We classify nonrational Fano threefolds X with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, (−KX)3≥8, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$. ...

Added: September 1, 2023

Iliev A., Katzarkov L., Victor Przyjalkowski, Proceedings of the Edinburgh Mathematical Society 2014 Vol. 57 P. 145-173

This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following M. Ballard, D. Favero, L. Katzarkov (ArXiv:1012.0864) and D. Favero, L. Katzarkov (Noether--Lefschetz Spectra and Algebraic cycles, in preparation) we enhance constructions from A. Kuznetsov (arXiv:0904.4330) by introducing Noether--Lefschetz spectra --- an interplay between Orlov spectra (C. Oliva, ...

Added: July 2, 2013

Golota A., / Cornell University. Series arXiv "math". 2022.

Let X be a complex projective variety. Suppose that the group of birational automorphisms of X contains finite subgroups isomorphic to (ℤ/Niℤ)r for r fixed and Ni arbitrarily large. We show that r does not exceed 2dim(X). We also show that the same result holds for groups of bimeromorphic automorphisms of compact Kähler threefolds. ...

Added: October 4, 2022

Vladimir L. Popov, Journal of the Ramanujan Mathematical Society 2013 Vol. 28A No. Special Issue-2013 dedicated to C.S.Seshadri's 80th birthday P. 409-415

We construct counterexamples to the rationality conjecture regarding the new version of the Makar-Limanov invariant formulated in A. Liendo, G_a-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653--3665. ...

Added: June 20, 2013

Kang M., Yuri Prokhorov, Journal of Algebra 2010 Vol. 324 No. 9 P. 2166-2197

Added: December 5, 2013

Kuznetsov A., Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic. ...

Added: August 19, 2020

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Andrey S. Trepalin, Central European Journal of Mathematics 2014 Vol. 12 No. 2 P. 229-239

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: December 3, 2013

Kuznetsov A., Prokhorov Y., Journal of the Institute of Mathematics of Jussieu 2022 P. 1-41

We prove rationality criteria over nonclosed fields of characteristic 00 for five out of six types of geometrically rational Fano threefolds of Picard number 11 and geometric Picard number bigger than 11 . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational varieties of this type. ...

Added: November 30, 2022

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428-466

Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...

Added: March 17, 2013

Poddiakov A., / Social Science Research Network. Series SSRN Working Paper Series "SSRN Working Paper Series". 2023.

An important and interesting phenomenon of the last few decades is the increasing number of mathematical studies of so-called intransitive dice with non-standard numbers on their faces and the popularization of them. The dice beat one another like in the rock-paper-scissors game. They violate the transitivity law (or axiom): “if it were true that whenever ...

Added: February 23, 2023

Trepalin A., Central European Journal of Mathematics 2014

Let $\bbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\bbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\bbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\bbk} / G$ is rational for an arbitrary ...

Added: October 14, 2013

Rumynin D., Colloquium Mathematicum 2021 Vol. 164 P. 123-131

We investigate geometry of D-affine varieties. Our main result is that a D-affine rational projective surface over an algebraically closed field is a generalised flag variety of a reductive group. ...

Added: September 7, 2021

Bogomolov F. A., Böhning C., Graf von Bothmer H., Central European Journal of Mathematics 2012 Vol. 10 No. 2 P. 466-520

Let G be one of the groups SL n(ℂ), Sp 2n(ℂ), SO m(ℂ), O m(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ N is rational. In this paper we improve known bounds for the levels of stable ...

Added: February 6, 2013

Васильев Д. А., Siberian Mathematical Journal 2023 Vol. 64 No. 3 P. 525-541

We construct an infinite series of irreducible components of the moduli space of stable rank 3 sheaves on P3 with the zero first Chern class and establish the rationality of the components of this series. We also prove the rationality of the irreducible components of the moduli space of stable rank 2 sheaves on P3 belonging to an infinite subseries of the series ...

Added: May 29, 2023

Prokhorov Y., Kuznetsov A., / Cornell University. Series arXiv "math". 2020.

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus g∈{7,8,9,10} over an arbitrary field 𝗄 of zero characteristic. In the case of dimension n≥4 we prove that these varieties are 𝗄-rational if and only if they have a 𝗄-point except for the case of genus 9, where we assume n≥5. Furthermore, we prove that Mukai varieties of ...

Added: August 19, 2020

Kuznetsov A., Prokhorov Y., American Journal of Mathematics 2023 Vol. 145 No. 2 P. 335-411

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic. ...

Added: September 1, 2023

Lubashevsky I., Wagner P., Mahnke R., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2003 Vol. 68 Article 056109

The problem of a car following a lead car driven with constant velocity is considered. To derive the governing equations for the following car dynamics a cost functional is constructed. This functional ranks the outcomes of different driving strategies, which applies to fairly general properties of the driver behavior. Assuming rational-driver behavior, the existence of ...

Added: November 5, 2021

Prokhorov Y., / Cornell University. Series arXiv "math". 2019.

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found. ...

Added: November 19, 2019

Prokhorov Y., Kuznetsov A., / Cornell University. Series arXiv "math". 2021.

We prove rationality criteria over algebraically non-closed fields of characteristic 0 for five out of six types of geometrically rational Fano threefolds of Picard number 1 and geometric Picard number bigger than 1. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions. ...

Added: November 23, 2021

Kuznetsov A., Perry A., Compositio Mathematica 2018 Vol. 154 No. 7 P. 1362-1406

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, ...

Added: September 13, 2018

Popov V., / Bielefeld University. Series LAGRS "Linear Algebraic Groups and Related Structures". 2012. No. 485.

We construct counterexamples to the rationality conjecture regar-ding the new version of the Makar-Limanov invariant introduced in A. Liendo, Ga-actions of fiber type on affine T-varieties, J. Algebra 324 (2010), 3653–3665. ...

Added: January 9, 2013